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Using Grothendieck Groups to Define and Relate the Hilbert and Chern Polynomials Axiomatically

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Using Grothendieck Groups to Define and Relate the Hilbert and Chern Polynomials Axiomatically USING GROTHENDIECK GROUPS TO DEFINE AND RELATE THE HILBERT AND CHERN POLYNOMIALS AXIOMATICALLY TAKUMI MURAYAMA Abstract. The Hilbert polynomial can be defined axiomatically as a group homo- morphism on the Grothendieck group K(X) of a projective variety X, satisfying certain properties. The Chern polynomial can be similarly defined. We there- fore define these rather abstract notions to try and find a nice description of this relationship. Introduction Let X = Pr be a complex projective variety over an algebraically closed field k with coordinate ring S = k[x0; : : : ; xr]. Recall that the Hilbert function of a finitely generated graded S-module M is defined as func hM (t) := dimk Mt; and that we have the following Theorem (Hilbert). There is a polynomial hM (t) such that func hM (t) = hM (t) for t 0: We call hM (t) the Hilbert polynomial of M. The Hilbert polynomial satisfies a nice property: given an exact sequence 0 −! M 0 −! M −! M 00 −! 0 of graded S-modules, where each homomorphism is graded, i.e., it respects the grading on the modules M; M 0;M 00, then (1) hM (t) = hM 0 (t) + hM 00 (t); by the fact that the sequence above is exact at each degree t, and by the rank-nullity theorem. The Hilbert polynomial therefore said to be \additive on exact sequences"; this is an extremely useful fact that enables us to compute the Hilbert polynomial in many cases. For example, we computed the following for homework: Example (HW8, #3). Let f(w; x; y; z) and g(w; x; y; z) be relatively prime nonzero homogeneous polynomials of degree a and b in S := k[w; x; y; z]. Then, ff; gg forms a Date: December 8, 2014. 1 2 TAKUMI MURAYAMA regular sequence, and so by [Eis95, Thm. 17.6], the Koszul sequence 0 g 1 @−f A f g 0 S(−a − b) S(−a) ⊕ S(−b) S S=(f; g) 0 is a graded free resolution for S=(f; g). By the additivity of the Hilbert polynomial, poly poly poly poly poly hS=(f;g)(t) = hS (t) − (hS (t − a) + hS (t − b)) + hS (t − a − b) t + 3 t + 3 − a t + 3 − b t + 3 − a − b = − − + : 3 3 3 3 This additivity property, however, is not unique to Hilbert polynomials. By a general construction in [FAC, no 57] and [Ser55, no 4], it is often1 possible to associate a vector bundle E to M. From a topological point of view (in which case we assume 2i k = C), to such a bundle we can associate Chern classes cE (i) 2 H (X; Z) = Z that are useful as an invariant of E . We can encode these Chern classes into a polynomial we call the Chern polynomial q X i cE = cE (i)x ; i=0 and an important formula used in computing Chern classes is the Whitney multiplica- tion formula, which says q+q0 q ! q0 ! X i X i X i (2) cE ⊕E 0 = cE ⊕E 0 (i)x = cE (i)x cE 0 (i)x = cE cE 0 ; i=0 i=0 i=0 which is found in, for example, [Hir95, Thm. 4.4.3] or [MS74, 14.7]. The Chern polynomial therefore transforms direct sums into products of polynomials. In this manner, Hilbert polynomials and Chern polynomials both behave \nicely" with respect to exact sequences, although the former is additive with respect to exact sequences, while the latter are multiplicative. It is goal in this note to define both axiomatically in the hope that this elucidates their relationship. We have striven to make most ideas as simple as possible, hopefully so that those who have taken a first semester course in commutative algebra following [AM69] or [Eis95] and in algebraic geometry following [Sha13] or [Har77, I], for example, will ideally be able to follow the arguments made. The exception is x2, which we regrettably were not able to make more elementary; we have instead tried to use [FAC] as our main source for cohomological results, since it does not require schemes. 1. Additive Functions and the Grothendieck Group Let C be an abelian category. The Hilbert and Chern polynomials are specific examples of the following general construction, following [SGA5, VIII 1]: 1In general, you only get a coherent sheaf. HILBERT AND CHERN POLYNOMIALS 3 Definition. A function λ on the objects of C with values in an abelian monoid G is additive if for all short exact sequences 0 −! M 0 −! M −! M 00 −! 0; in C , where M 0;M 00;M are objects in C , we have λ(M) = λ(M 0) + λ(M 00): Example 1. We explain here why exactly the Hilbert and Chern polynomials are examples of this construction. Let S and X as in the Introduction, where we recall X = Pr. Let gr-S be the category of finitely generated graded S-modules. Then, the function M 7! hM (t) 2 Q[t] is additive by (1). Now let vect-X be the category of vector bundles2 over X. Then, the function r+1 E 7! cE (t) 2 Z[t]=(t ) is multiplicative by (2). Note that it is therefore additive if we consider Z[t]=(tr+1) as an abelian monoid under multiplication. Now let F (C ) denote the free abelian group generated by isomorphism classes of objects in C . Let Q(C ) be the subgroup of F (C ) generated by formal sums of the form [M 0] − [M] + [M 00] for each short exact sequence 0 −! M 0 −! M −! M 00 −! 0 in C , where [M] is the isomorphism class of M. We then have the following Definition. The quotient group F (C )=Q(C ) is called the Grothendieck group of C , and is denoted K(C ). If M is an object of C , then denote γ(M) to be the image of [M] in K(C ). Now if the abelian monoid G is actually a group, the Grothendieck group K(C ) is characterized by the following Universal Property. For each additive function λ on C , with values in an abelian group G, there exits a unique homomorphism λ0 : K(C ) ! G such that λ(M) = λ0(γ(M)) for all M. Proof. This follows from the universal property for quotient groups, which gives us a unique group homomorphism λ0 fitting into the diagram F (C ) λ G λ0 K(C ) The study of additive functions on a category C with values in an abelian group G, then, can be reduced to the study of group homomorphisms K(C ) ! G. We compute some elementary examples of Grothendieck groups: 2Technically we only defined additive functions on abelian categories. However, the same definition goes through for non-abelian categories, as long as there is a good notion of an exact sequence. 4 TAKUMI MURAYAMA Example 2 (Modules over a Noetherian ring, cf. [AM69, Exc. 7.26]). Let A be a Noetherian ring, and let K(mod-A) be the Grothendieck group of mod-A, the category of finitely generated A-modules. By d´evissage [AM69, Exc. 7.18], for each finitely generated A-module M, there exists a chain of submodules 0 = M0 ( M1 ( ··· ( Mr = M; ∼ such that Mi=Mi−1 = A=pi for each i, where pi ⊂ A are prime ideals. We then have a short exact sequence 0 −! Mr−1 −! M −! M=Mr−1 −! 0; so γ(M) = γ(Mr−1) + γ(M=Mr−1) = γ(Mr−1) + γ(A=pr) in K(mod-A). By induction, r X γ(M) = γ(A=pi); i=1 and so K(mod-A) is generated by γ(A=p) for each prime ideal p ⊂ A. In particular, if A is a PID and M is a finitely-generated A-module, then by the structure theorem for finitely generated modules over a PID, ∼ ⊕r M = A ⊕ A=(f1) ⊕ A=(f2) ⊕ · · · ⊕ A=(fn) where fi j fi+1 for all i. Since we have the short exact sequence ·fi 0 −! A −! A −! A=(fi) −! 0; r we see γ(A=(fi)) = γ(A) − γ(A) = 0. Thus, γ(M) = γ(A ), and since γ is additive by the fact that the short exact sequence 0 −! M −! M ⊕ M 0 −! M 0 −! 0 gives the relation γ(M ⊕ M 0) = γ(M) + γ(M 0), we have that γ is additive and moreover γ(M) = r · γ(A). Thus, K(mod-A) is the abelian group generated by γ(A), ∼ and mapping γ(A) to 1 gives an isomorphism K(mod-A) = Z since the order of γ(A) is infinite. Example 3 (Graded modules over a Noetherian graded ring). Let S be a Noetherian graded ring, and let K(gr-S) be the Grothendieck group of the category of finitely generated graded S-modules. As in the previous example, we can apply d´evissage [Har77, I, Prop. 7.4]: for each finitely generated graded S-module M, there exists a chain of graded submodules 0 1 r 0 = M ( M ( ··· ( M = M; i i−1 ∼ such that M =M = (S=pi)(li) for each i, where pi ⊂ S are homogeneous prime ideals, and li 2 Z denotes a degree shift M(l)t = Mt+l. Just like in the previous example, r X γ(M) = γ((S=pi)(li)); i=1 HILBERT AND CHERN POLYNOMIALS 5 and so K(gr-S) is generated by γ((S=p)(l)) for each homogeneous prime ideal p ⊂ S and grade shifts l 2 Z. Also in analogy with the previous example, it is possible, with more work, to show ∼ that if S = k[x; y], then K(gr-S) = Z ⊕ Z.
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